Minimizing Norms to Solve Approximately a Frequency Assignment Problem

We show that solving the frequency assignment problem is equivalent to solving a minimization problem involving spectral norms of matrices, if the link gain matrix is symmetric. Often the spectral norm is ideal for minimization problems, but not so in this case. We reformulate the minimization using the one-, innnity-and Frobenius norm. This allows us to deal with non symmetric link gain matrices as well. We propose two types of algorithms to solve these new minimization problems approximately. The algorithms minimizing the one-and/or innnity-norm are exchanging rows and/or columns based on the weights of the elements therein whereas the algorithm minimizing the Frobenius norm is derived from techniques in spectral bisection of graphs. Numerical results show that for regular distributions of base stations the algorithm minimizing the Frobenius norm computes good approximations, whereas for irregular distributions of base stations the algorithms minimizing the one-and/or innnity-norm are far superior .